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The function in example 1, a removable discontinuity. Consider the piecewise function = {< = >. The point = is a removable discontinuity.For this kind of discontinuity: The one-sided limit from the negative direction: = and the one-sided limit from the positive direction: + = + at both exist, are finite, and are equal to = = +.
In other words, the function has an infinite discontinuity when its graph has a vertical asymptote. An essential singularity is a term borrowed from complex analysis (see below). This is the case when either one or the other limits f ( c − ) {\displaystyle f(c^{-})} or f ( c + ) {\displaystyle f(c^{+})} does not exist, but not because it is ...
A graph of a parabola with a removable singularity at x = 2 In complex analysis , a removable singularity of a holomorphic function is a point at which the function is undefined , but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourhood of that point.
Since the value at f(0) is a removable discontinuity, = for all a. Thus, the naïve chain rule would suggest that the limit of f ( f ( x )) is 0. However, it is the case that f ( f ( x ) ) = { 1 if x ≠ 0 0 if x = 0 {\displaystyle f(f(x))={\begin{cases}1&{\text{if }}x\neq 0\\0&{\text{if }}x=0\end{cases}}} and so lim x → a f ( f ( x ) ) = 1 ...
The field of asymptotics is normally first encountered in school geometry with the introduction of the asymptote, a line to which a curve tends at infinity. The word Ασύμπτωτος (asymptotos) in Greek means non-coincident and puts strong emphasis on the point that approximation does not turn into coincidence.
Then f is a non-decreasing function on [a, b], which is continuous except for jump discontinuities at x n for n ≥ 1. In the case of finitely many jump discontinuities, f is a step function. The examples above are generalised step functions; they are very special cases of what are called jump functions or saltus-functions. [8] [9]
A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist.
The next example shows that, computing a residue by series expansion, a major role is played by the Lagrange inversion theorem. Let u ( z ) := ∑ k ≥ 1 u k z k {\displaystyle u(z):=\sum _{k\geq 1}u_{k}z^{k}} be an entire function , and let v ( z ) := ∑ k ≥ 1 v k z k {\displaystyle v(z):=\sum _{k\geq 1}v_{k}z^{k}} with positive radius of ...